Hilbert's $19^{\text{th}}$ problem revisited

TL;DR

This survey revisits Hilbert's 19th problem, reviewing classical regularity results for minimizers and recent advances in degenerate convex functionals, highlighting open problems and research directions.

Contribution

It provides a comprehensive overview of the classical and recent developments in the regularity theory related to Hilbert's 19th problem.

Findings

Classical regularity results for minimizers in all dimensions

Recent progress on degenerate convex functionals

Identification of open problems in the field

Abstract

In this survey article we revisit Hilbert's $19^{\text{th}}$ problem concerning the regularity of minimizers of variational integrals. We first discuss the classical theory (that is, the statement and resolution of Hilbert's problem in all dimensions). We then discuss recent results concerning the regularity of minimizers of degenerate convex functionals. Finally, we discuss some open problems. Exercises are included for the benefit of researchers who are entering the subject.